An analogous equation also holds for .
This means that for a
closed shell RHF reference,
(
)
for singlets,
and
(
)
for triplets; thus only half
of the CI coefficients must be computed explicitly. These sign rules
are also evident from the observation that
and
are not spin
eigenfunctions, but that the total CI wavefunction will be a spin
eigenfunction (if the required determinants are present in the CI
space). Using the determinant sign convention of Szabo and Ostlund
[6], spin eigenfunctions (or CSFs) associated with
the above determinants are

=

(27)

=

(28)

Using these relationships between CI coefficients, we obtain

=

(29)

=

(30)

Furthermore,
and
.

Consider how equations
(20)-(22) change when spin is
explicitly accounted for. There will be two pseudodensity matrices,

=

(31)

=

(32)

Due to equation (26),
for singlets,
and
for triplets. Thus it is necessary to form only one
of the Fock-like matrices,
or
.
The former is constructed as

=

(33)

=

(34)

Finally
is constructed according to eq
(22). In terms of these quantities, the
vector can be written